hilbert-spaces

# Hilbert Spaces ## When to Use Use this skill when working on hilbert-spaces problems in functional analysis. ## Decision Tree 1. **Orthogonal decomposition** - For closed subspace M: H = M + M^perp (direct sum) - Every x = P_M(x) + P_{M^perp}(x) - `sympy_compute.py simplify "x - projection"` 2. **Projection Theorem** - For closed convex C, unique nearest point exists - P_C is nonexpansive: ||P_C(x) - P_C(y)|| <= ||x - y|| - `z3_solve.py prove "projection_exists_unique"` 3. **Riesz Representation** - Every f in H* has form f(x) = <x, y_f> for unique y_f - ||f|| = ||y_f|| - `z3_solve.py prove "riesz_representation"` 4. **Parseval's Identity** - For orthonormal basis {e_n}: ||x||^2 = sum|<x, e_n>|^2 - `sympy_compute.py sum "abs(<x, e_n>)**2"` 5. **Bessel's Inequality** - sum|<x, e_n>|^2 <= ||x||^2 for any orthonormal set ## Tool Commands ### Sympy_Inner_Product ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "<x + y, z> == <x,z> + <y,z>" ``` ### Z3_Projection ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "x - P_M(x) in M_perp" ``` ### Z3_Riesz ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "bounded_linear_functional iff inner_product_form" ``` ### Sympy_Parseval ```bash uv run python -m runtime.harness scripts/sympy_compute.py sum "abs(<x, e_n>)**2" --var n --from 1 --to oo ``` ## Key Techniques *From i...